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I have a matrix, $X'X$, which is singular meaning that I cannot invert it. I need the inverse of this matrix to perform two independent things. I need it for the design of experiments, in R using optFederov() algorithm. Also I need it invertible for OLS.

I believe the two possible sources of singularity are:

1.- Many points lying in the same plane

2.- There are linear combinations between columns

I think it is the first problem. To solve the problem I have added very small random noise to each value of x, $x_{i}$, from a Normal Distribution such that $x_{i,noise} \sim N(0,0.001)$.

Apparently this trick makes the matrix invertible. However, I am not sure if this is a good approach. My thought is that it is good for OLS given that the esitmators will change veeery little, but not sure on the design of experiments, for example using a D-Efficiency design.

I would appreciate some suggestion regarding whether this is a good way to proceed or not!

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  • $\begingroup$ Presumably $X$ is the design matrix. $\endgroup$ Commented Mar 30, 2017 at 8:30
  • $\begingroup$ @Scortchi yes, X is the full factorial design, but in order to obtain a D-optimal fractional factorial design I have to minimize $|(X'X)^{-1}|$ which indeed requires the inverse of $X'X$ $\endgroup$
    – adrian1121
    Commented Mar 30, 2017 at 8:36
  • $\begingroup$ en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse ? $\endgroup$ Commented Mar 30, 2017 at 11:23
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    $\begingroup$ If it is singular certainly it cannot be Doptimal, unless tjere are so few points that all designs are singular. You shpuld splve that problem rather than just perturb $\endgroup$ Commented Mar 30, 2017 at 13:49

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Rather than random noise, adding a small multiple of the identity matrix is the standard way to handle this in regression problems. It's equivalent to ridge regression. I.e $X^\prime X+\epsilon I$ is used in place of $X^\prime X$, for $\epsilon$ small, say 0.001 or 0.01. I'm not sure about the experimental design aspects.

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  • $\begingroup$ Yes, but it wouldbebetter tofind out what whent wrong $\endgroup$ Commented Mar 30, 2017 at 13:51
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    $\begingroup$ This is a good point, but it would be well to explain that "$\epsilon$ small" is meaningful only when the columns of the design matrix have been standardized. $\endgroup$
    – whuber
    Commented Mar 30, 2017 at 14:21

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